基于Copula密度函数的毛竹胸径年龄结构特征二元联合分布

doi:10.11707/j.1001-7488.20211110

Abstract

Abstract:

Objective: In view of the shortcomings of the bivariate joint distribution function commonly used in the investigation of forest structure characteristic factors, a bivariate distribution(density)function with a low applicable condition, wide adaptation range and great application value was selected to provide a reference for accurately measuring the joint distribution of forest structure characteristic factors. Method: Five commonly used bivariate Copula density functions, bivariate S_bb functions and bivariate Weibull distribution functions, were selected to describe the bivariate joint density of DBH and age of moso bamboo(Phyllostachys edulis) based on 177 continuous inventory plots in Zhejiang Province in 2009, and the measurement accuracy of each function was compared and analyzed. The optimal Copula function of DBH and age was selected based on the metrics of bivariate frequency histogram and AIC. The bivariate Copula joint density model of DBH and age for moso bamboo in Zhejiang Province was established. The goodness of fit of the model was tested using Kolmogorov test. Result: The coefficient of determination(R²) of bivariate Weibull distribution function and the bivariate S_bb function was 0.990 1 and 0.736 2, respectively, and the R² of bivariate Gumbel Copula density function was 0.984 1 with the lowest AIC value of -19.519 6. The maximum deviations of the cumulative value of the bivariate Gumbel Copula function, the bivariate Weibull distribution function and the bivariate S_bb function were 0.007 0, 0.015 8 and 0.078 1. The critical significance value was 0.179 8. Conclusion: Bivariate Gumbel Copula probability density function is the best Copula function for representing the joint distribution of moso bamboo DBH and age. The measurement accuracy of bivariate Weibull distribution function is the highest, but the number of the parameters used is more than that of those used in the bivariate Copula function. Therefore, iterative parameter of bivariate Copula function is easier to be converged. The accuracy of bivariate S_bb function is the lowest. The joint distribution of DBH and age of moso bamboo all obey these three distribution functions. Bivariate Copula function is suitable for arbitrary edge distribution of DBH and age, that is, it does not need to determine the class of marginal distribution function. When there are only marginal distribution values of DBH and age, the joint density value of DBH and age could be obtained using the bivariate Copula function. Thus, the bivariate Copula function has a wider applicability and a higher application value than the commonly used binary distribution function.

Key words: moso bamboo, bivariate distribution, Copula density function, biomass, structure characteristics of DBH and age

CLC Number:

S758

Enbin Liu,Hongwen Yao,Zexi Ren,Guomo Zhou,Huaqiang Du. Bivariate Joint Distribution of DBH and Age of Moso Bamboo Based on Copula Density Function[J]. Scientia Silvae Sinicae, 2021, 57(11): 94-104.

Figures/Tables 8

Table 1

Fig.1

Fig.2

Table 2

The measured probability and estimated values of the bivariate Gumbel Copula density function, the bivariate Weibull distribution function and the bivariate Sbb density function"

胸径 DBH/cm	1度1du				2度2du
胸径 DBH/cm	实测概率 Measured probability	GumCp估计值 GumCp estimatedvalues	S_bb估计值 S_bb estimatedvalues	Weibull估计值 Weibull estimated values	实测概率 Measured probability	GumCp估计值 GumCp estimated values	S_bb估计值 S_bb estimated values	Weibull估计值 Weibull estimated values
5	0.014 1	0.016 6	0.017 2	0.015 2	0.031 5	0.029 6	0.016 1	0.031 2
6	0.015 1	0.017 5	0.031 0	0.015 5	0.034 2	0.031 9	0.029 0	0.031 0
7	0.022 9	0.024 1	0.041 6	0.022 8	0.048 2	0.045 2	0.039 1	0.044 7
8	0.028 0	0.027 9	0.046 2	0.028 4	0.055 5	0.053 7	0.043 4	0.055 2
9	0.028 7	0.027 7	0.044 2	0.029 5	0.055 1	0.054 8	0.041 6	0.057 1
10	0.028 0	0.024 9	0.036 9	0.025 0	0.048 8	0.050 8	0.034 8	0.048 4
11	0.016 2	0.014 2	0.026 7	0.016 6	0.026 1	0.029 4	0.025 2	0.032 6
12	0.009 7	0.007 3	0.016 3	0.008 3	0.013 9	0.015 4	0.015 4	0.016 7
13	0.002 9	0.002 3	0.007 9	0.003 0	0.004 6	0.004 8	0.007 4	0.006 2
14	0.000 7	0.000 4	0.002 6	0.000 7	0.001 0	0.000 9	0.002 5	0.001 6
15	0	0	0.000 4	1E-04	0	0.000 1	0.000 4	0.000 3

胸径 DBH/cm	3度3du				4度4du
胸径 DBH/cm	实测概率 Measured probability	GumCp估计值 GumCp estimated values	S_bb估计值 S_bb estimated values	Weibull估计值 Weibull estimated values	实测概率 Measured probability	GumCp估计值 GumCp estimated values	S_bb估计值 S_bb estimated values	Weibull估计值 Weibull estimated values
5	0.032 1	0.028 4	0.014 9	0.032 2	0.012 3	0.011 0	0.013 3	0.014 3
6	0.032 4	0.031 2	0.026 9	0.032 0	0.014 4	0.012 1	0.024 1	0.014 2
7	0.045 8	0.045 2	0.036 3	0.046 4	0.018 3	0.017 8	0.032 6	0.020 5
8	0.056 7	0.055 6	0.040 4	0.057 0	0.021 1	0.022 2	0.036 4	0.025 3
9	0.057 5	0.059 4	0.038 8	0.059 1	0.024 6	0.024 6	0.034 9	0.026 2
10	0.053 2	0.059 2	0.032 4	0.050 1	0.026 6	0.026 9	0.029 3	0.022 2
11	0.034 2	0.036 9	0.023 5	0.033 7	0.017 5	0.020 3	0.021 3	0.014 9
12	0.019 0	0.020 4	0.014 4	0.017 3	0.009 9	0.016 1	0.013 1	0.007 7
13	0.006 4	0.006 5	0.007 0	0.006 5	0.004 1	0.006 9	0.006 3	0.002 9
14	0.001 5	0.001 3	0.002 3	0.001 7	0.000 7	0.001 5	0.002 1	0.000 7
15	0.000 2	0.000 1	0.000 4	0.000 3	0.000 1	0.000 1	0.000 3	1E-04

胸径 DBH/cm	5度5du
胸径 DBH/cm	实测概率 Measured probability	GumCp估计值 GumCp estimated values	S_bb估计值 S_bb estimated values	Weibull估计值 Weibull estimated values
5	0.001 5	0.001 5	0.001 4	0.002 3
6	0.002 2	0.001 7	0.002 6	0.002 2
7	0.004 0	0.002 5	0.003 6	0.003 3
8	0.004 2	0.003 1	0.004 2	0.004 0
9	0.004 2	0.003 5	0.004 1	0.004 0
10	0.004 6	0.003 9	0.003 6	0.003 5
11	0.002 9	0.003 2	0.002 7	0.002 4
12	0.001 6	0.003 8	0.001 7	0.001 2
13	0.000 5	0.005 2	0.000 8	0.000 4
14	0.000 3	0.007 0	0.000 3	0.000 2
15	0	0.000 6	0.000 1	0

Table 2

Fig.3

Fig.4

Fig.5

Fig.6

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胸径 DBH/cm	1度1du		2度2du		3度3du		4度4du		5度5du
胸径 DBH/cm	株数 Stem number/hm^-2	累计株数 Cumulative stem	株数 Stem number/hm^-2	累计株数 Cumulative stem	株数 Stem number/hm^-2	累计株数 Cumulative stem	株数 Stem number/hm^-2	累计株数 Cumulative stem	株数 Stem number/hm^-2	累计株数 Cumulative stem
5	529	529	1 182	1 711	1 203	2 914	461	3 375	57	3 432
6	565	1 094	1 283	3 559	1 215	5 977	541	6 979	84	7 120
7	858	1 952	1 804	6 221	1 716	10 355	686	12 043	148	12 332
8	1 048	3 000	2 081	9 350	2 123	15 607	790	18 085	157	18 531
9	1 074	4 074	2 066	12 490	2 156	20 903	922	24 303	158	24 907
10	1 050	5 124	1 829	15 369	1 995	25 777	996	30 173	173	30 950
11	608	5 732	978	16 955	1 283	28 646	655	33 697	108	34 582
12	363	6 095	520	17 838	712	30 241	370	35 662	61	36 608
13	109	6 204	171	18 118	240	30 761	152	36 334	20	37 300
14	27	6 231	37	18 182	58	30 883	25	36 481	10	37 457
15	1	6 232	0	18 183	6	30 890	2	36 490	0	37 466

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Bivariate Joint Distribution of DBH and Age of Moso Bamboo Based on Copula Density Function

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